Optimizing the Crisp Representation of the Fuzzy...

Optimizing the Crisp Representation of theFuzzy Description Logic SROIQ

Fernando BobilloJoint research with Miguel Delgado and Juan Gomez-Romero

Department of Computer Science and Artificial IntelligenceUniversity of Granada, Spain

URSW 2007Busan (South Korea), November 2007

F. Bobillo (DECSAI, UGR) Optimized Crisp Representation of SROIQ URSW 2007 1 / 22

Outline

1 Motivation

2 Fuzzy SROIQ

3 A Crisp Representation for Fuzzy SROIQ

4 Implementation

5 Conclusions and Future Work


Crisp Representations for Fuzzy DLs

Classical ontologies are not appropriate for imprecise and vagueknowledge. A solution are fuzzy Description Logics (DLs).Fuzzy DLs require that new languages need to be used, andhence to adapt the available resources.

Specially difficult with reasoners: significant gap between thedesign of a decision procedure and a practical implementation.

Alternative: To represent fuzzy DLs using crisp DLs and toreduce reasoning within fuzzy DLs to reasoning within crisp ones.Advantages:

No need to agree a new standard fuzzy language.Use of standard languages and reuse of available resources.Use of existing crisp reasoners.

This will support early reasoning in future fuzzy languages.

An immediate practical application of fuzzy ontologies is feasible,because it relies on existing valid languages and tools.


Our contributions

A crisp representation of fuzzy SROIQ.G. Stoilos et al. proposed fuzzy SROIQ but only providedreasoning for a fragment of it, missing:

Qualified cardinality restrictions e.g. ≥ 2hasSon.Male,Negated local reflexivity concepts e.g. ¬∃likes.Self ,Negative role assertions e.g. (fernando, juan) : ¬hasFriend ,

Fuzzy Concept and Role Inclusion Axioms (GCIs and RIAs) canbe true to some degree using Godel implication in the semantics.

First work supporting reasoning with fuzzy RIAs.

The reduction is optimized in several ways:We reduce the number of new crisp atomic elements (which areneeded to represent the elements in the fuzzy KB).We reduce the new axioms needed to preserve their semantics.We show how to optimize some important GCIs.

Implementation of DELOREAN, the first reasoner supporting afuzzy extension of SHOIN (and hence OWL DL).


Outline

1 Motivation

2 Fuzzy SROIQ


4 Implementation



Complex fuzzy concept and roles

Constructor Syntax Semantics

(top concept) > 1

(bottom concept) ⊥ 0

(atomic concept) A AI (a)

(concept conjunction) C u D CI (a)⊗ DI (a)

(concept disjunction) C t D CI (a)⊕ DI (a)

(concept negation) ¬C CI (a)

(universal quantification) ∀R.C infb∈∆I {RI (a, b) → CI (b)}

(existential quantification) ∃R.C supb∈∆I {RI (a, b)⊗ CI (b)}

(fuzzy nominals)Sm

i=1{αi /oi} supi | a∈{oIi }αi

(at-least restriction) ≥ n S.C supb1,...,bm∈∆I [(⊗mi=1{SI (a, bi )⊗ CI (bi )})

N(⊗j<k{bj 6= bk})]

(at-most restriction) ≤ n S.C infb1,...,bn+1∈∆I [(⊗n+1i=1 {SI (a, bi )⊗ CI (bi )}) → (⊕j<k{bj = bk})]

(local reflexivity) ∃S.Self SI (a, a)

(atomic role) RA RIA (a, b)

(inverse role) R− RI (b, a)

(universal role) U 1


Fuzzy axioms in the ABox, TBox and RBox

Axiom Syntax Semantics

(concept ass.) 〈a :C ./ α〉 CI (aI ) ./ α

(role ass.) 〈(a, b) :R ./ α〉 RI (aI , bI ) ./ α

(inequality ass.) 〈a 6= b〉 aI 6= bI

(equality ass.) 〈a = b〉 aI = bI

(GCI) 〈C v D B α〉 infa∈∆I {CI (a) → DI (a)}B α

(RIA) 〈R1R2 . . . Rn v R′ B α〉 supb1...bn+1∈∆IN

[RI1 (b1, b2), ..., RIn (bn, bn+1)] → RI (b1, bn+1) B α

(transitive) trans(R) ∀a, b ∈ ∆I , RI (a, b) ≥ supc∈∆I RI (a, c)⊗ RI (c, b)

(disjoint) dis(S1, S2) ∀a, b ∈ ∆I , SI1 (a, b)⊗ SI2 (a, b) = 0

(reflexive) ref (R) ∀a ∈ ∆I , RI (a, a) = 1

(irreflexive) irr(S) ∀a ∈ ∆I , SI (a, a) = 0

(symmetric) sym(R) ∀a, b ∈ ∆I , RI (a, b) = RI (b, a)

(asymmetric) asy(S) ∀a, b ∈ ∆I , if SI (a, b) > 0 then SI (b, a) = 0


Fuzzy operators

We consider fKDSROIQ:

Minimum t-norm, α⊗ β = min{α, β}Maximum t-conorm, α⊕ β = max{α, β}Łukasiewicz negation, α = 1− α

KD implication except in GCIs and RIAs: α → β = max{1− α, β}

Godel implication in GCIs and RIAs: α → β =

{1 α ≤ ββ α > β

These fuzzy operators make possible the reduction to a crisp KB(other fuzzy operators are not suitable in principle).


Fuzzy GCIs and RIAs

The most common semantics for GCIs and RIAs, based onZadeh’s set inclusion, forces them to be either true or false:

C v D iff ∀x ∈ ∆I , CI(x) ≤ DI(x)

The use of Kleene-Dienes implication in the semantics of GCIsand RIAs brings about two counter-intuitive effects:

In general concepts (and roles) do not fully subsume themselves.〈C v D ≥ 1〉 force some fuzzy concepts and roles to be interpretedas crisp.

Godel implication:

Solves these problems,It is suitable for a classical representation.For GCIs of the form 〈C v D ≥ 1〉, it is equivalent to considerZadeh’s set inclusion.


Outline

1 Motivation

2 Fuzzy SROIQ


4 Implementation



Idea of the reduction

1 Compute the set of degrees of truth which must be considered.Degrees x in the fuzzy KB and their complementaries 1− x .

2 For each fuzzy atomic concept and role, add new crisp elements(α-cut and strict α-cut.)

3 Add new axioms to preserve their semantics.4 Reduce fuzzy axioms using the new crisp elements.

The size of the crisp KB is quadratic.The size is linear under a fixed set of degrees.

The reduction preserves reasoning.Consistency of the fuzzy KB and the crisp KB are equivalent.

The reduction can be reused when adding new axioms.If the new axiom do not introduce new vocabulary nor degrees.


Example of reduction

h is a hotel at a German-speaking country with at-least degree 0.5.KB = { 〈h : Hotel u ∀isIn.{(ge, 1), (au, 1), (sw , 0.67)} > 0.5〉 }

1 Degrees of truth to be considered:{0, 0.5, 1}

2 New crisp elements:Hotel>0, Hotel≥0.5, Hotel>0.5, Hotel≥1, isIn>0, isIn≥0.5, isIn>0.5, isIn≥1

3 New axioms:Hotel≥0.25 v Hotel>0, Hotel>0.25 v Hotel≥0.25, . . .

IsIn≥0.25 v IsIn>0, IsIn>0.25 v IsIn≥0.25, . . .

4 Reduction of every axiom in the KB:h : ρ(Hotel , > 0.5) u ∀ρ(isIn,≥ 0.5).ρ({(ge, 1), (au, 1), (sw , 0.67)},> 0.5) = h : Hotel>0.5 u ∀isIn≥0.5.{ge, au, sw}


Example: Reduction of a fuzzy GCI

Consider the GCI 〈C v D ≥ α〉.If it is satisfied, infa∈∆I CI(a) ⇒ DI(a) ≥ α.An arbitrary a must satisfy that CI(a) ⇒ DI(a) ≥ α.From the semantics of Godel implication, this is true if:

CI(a) ≤ DI(a), orDI(a) ≥ α.

Roughly, for very γ such that γ < α, CI(a) B γ implies DI(a) B γ.

ρ(C,Bγ) v ρ(D,Bγ)

Additionally, CI(a) ≥ α implies DI(a) ≥ α.

ρ(C,≥ α) v ρ(D,≥ α)


Optimizing the number of new elements and axioms

Roughly, for each α, β ∈ N fK we create:Two new crisp atomic concepts A≥α, A>β .Two new crisp atomic roles R≥α, R>β .

Previous works use two more atomic concepts A≤β, A<α, but:We use ¬A>γk rather than A≤γk .We use ¬A≥γk instead of A<γk .

We also need some new axioms to preserve their semantics:

A≥γi+1 v A>γi A>γj v A≥γj

R≥γi+1 v R>γi R>γi v R≥γi

Previous works also use some additional axioms, which noware superfluous (they follow immediately from the semantics):

A<γk v A≤γk A≤γi v A<γi+1

A≥γk u A<γk v ⊥ A>γi u A≤γi v ⊥> v A≥γk t A<γk > v A>γi t A≤γi


Optimizing some GCIs

〈C v > ./ γ〉 and 〈⊥ v D ./ γ〉 are tautologies.They are unnecessary in the resulting KB.

κ(> v D ./ γ) = > v ρ(D, ./ γ).

It appears in role range axioms, range(R) = C iff > v ∀R.C ≥ 1.

κ(C v ⊥ ./ γ) = ρ(C, > 0) v ⊥.

It appears in disjointness, disjoint(C, D) = C iff C u D v ⊥ ≥ 1.

If the resulting TBox contains A v B, A v C and B v C, thenA v C is unnecessary.

Example: κ(C v {1/o1, 0.5/o2}) ={C>0 v {o1, o2}, C≥0.5 v {o1, o2}, C>0.5 v {o1}, C≥1 v {o1}}can be optimized to: {C>0 v {o1, o2}, C≥0.5 v {o1}}.


Outline

1 Motivation

2 Fuzzy SROIQ


4 Implementation



Implementation: DELOREAN

DELOREAN = DEscription LOgic REasoner with vAgueNess.Using Java, Jena API, JavaCC and DIG 1.1 interface.Architecture:

The Parser reads an input file with a fuzzy ontology.Reduction module implements the reduction, builds a Jena model,and saves it as an OWL file with an equivalent crisp ontology.Inference module perform a consistency test, using any crispreasoner through the DIG interface.User interface communicates with the user.

Currently the logic supported is fKDSHOIN (OWL DL), sinceDIG interface does not yet support full SROIQ.


DELOREAN User Interface


Experimentation

Experiments have shown that the results of the reasoning tasksover the crisp ontology were the expected.We extended the axioms of Koala, a small ALCON (D) ontology,with random degrees and used PELLET reasoner through DIG.Time of a classification test over the resulting crisp ontology:

Number of degrees crisp 3 5 7 11Reduction time - 1.18 6.28 23.5 148.25Reasoning time 0.56 0.98 1.343 2.88 6.47

The reduction time is currently high, so the implementationshould be optimized. Anyway, the reduction can be reused andhence needs to be computed just once (possibly off-line).The reasoning time is reasonable at least for small ontologiesand using a limited number of degrees of truth.


Outline

1 Motivation

2 Fuzzy SROIQ


4 Implementation



Conclusions and Future Work

Conclusions:

We have shown how to reduce fuzzy SROIQ into SROIQ.Crisp representations can be optimized in several ways.Restricting the number of truth degrees is important to control thecomplexity of the reduction.DELOREAN is the first reasoner supporting fuzzy SHOIN (andhence fuzzy OWL DL).

Future work:

Compare DELOREAN with other fuzzy DL reasoners.Extend the reasoner to fuzzy SROIQ (and hence OWL 1.1) assoon as DIG 2.0 interface is available.To allow the definition of crisp concepts and roles.To allow the use of two implications in the semantics of GCIs andRIAs: Godel and Kleene-Dienes.


Questions?

Thank you very much for your attention


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